Problem: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$5.50$ each for teachers and $$3.50$ each for students, and the group paid $$46.00$ in total. The next month, the same group visited a science museum where the tickets cost $$16.50$ each for teachers and $$10.00$ each for students, and the group paid $$133.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+3.5y = 46}$ ${16.5x+10y = 133}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-10.5y = -138}$ ${16.5x+10y = 133}$ Add the top and bottom equations together. $ -0.5y = -5 $ $ y = \dfrac{-5}{-0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5.5x+3.5y = 46}$ to find $x$ ${5.5x + 3.5}{(10)}{= 46}$ $5.5x+35 = 46$ $5.5x = 11$ $x = \dfrac{11}{5.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {16.5x+10y = 133}$ and get the same answer for $x$ ${16.5x + 10}{(10)}{= 133}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.